A generalization of reduced rings. (Q2909800)

Let \(R\) be a ring with \(1\), \(\delta\) a derivation of \(R\) and \(R[x;\delta]\) the differential polynomial ring with the usual addition and multiplication such that \(xa=ax+\delta(a)\) for all \(a\in R\). Then \(R\) is called a \(\delta\)-Armendariz ring if for each \(f(x)=\sum_{i=0}^na_ix^i\), \(g(x)=\sum_{j=0}^mb_jx^j\in R[x;\delta]\), \(f(x)g(x)=0\) implies \(a_i\delta^i(b_j)=0\) for all \(i, j\). A ring \(R\) is called weak \(\delta\)-Armendariz if the above condition holds for \(n=m=1\). Then the authors show some properties of the radicals of \(R\) and relations between \(R\) and \(R[x;\delta]\) for a weak \(\delta\)-Armendariz ring \(R\).NEWLINENEWLINE Theorem 1. Let \(N_0(R)\), \(\text{Nil}_*(R)\), \(\text{L-rad}(R)\), \(\text{Nil}^*(R)\) be the Wedderburn, lower nil, Levitzky, and upper nil radical, respectively. If \(R\) is weak \(\delta\)-Armendariz, thenNEWLINENEWLINE (1) the above radicals are equal; andNEWLINENEWLINE (2) let \(J(R[x;\delta])\) be the Jacobson radical of \(R[x;\delta]\). Then \(J(R[x;\delta])\cap R\) is a nil ideal of \(R\).NEWLINENEWLINE If \(R\) is \(\delta\)-Armendariz, then the above radicals of \(R[x;\delta]\) are equal.NEWLINENEWLINE Theorem 2. Let \(R\) be a \(\delta\)-Armendariz ring. Then \(R\) is reversible (resp. symmetric, \(\delta\)-quasi Baer, \(\delta\)-Baer, p.p.-Baer) if and only if so is \(R[x;\delta]\) (resp. symmetric, \(\delta\)-quasi Baer, \(\delta\)-Baer, p.p.-Baer). Moreover, a reduced ring \(R\) with a \(\delta\) is characterized in terms of some differential Armendariz matrix rings. Also an Ore ring \(R\) with a \(\delta\) which is a (weak) \(\delta\)-Armendariz ring is characterized in terms of the (weak) differential classical left quotient ring of \(R\) where the derivation is induced by \(\delta\) of \(R\).